#506086

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,33)(20,37)(21,40)(22,30)(23,29)(24,31)(25,44)(26,34)(27,36)(28,35)(32,42)(38,48)(39,47)(41,46)(43,45);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,35,25,21,38,20,24,36,43,39,42,41,22,34)(18,26,30,46,32,47,45,27,31,37,48,40,44,28,29,33);(11,15)(13,16)(17,20,38,41,42,35,23,36,24,34,22,21,25,39,43,19)(18,27,45,28,44,46,30,37,31,33,29,47,32,40,48,26);(1,3,4,8)(2,6,5,7)(17,21,23,20,25,36,38,39,24,41,43,34,42,19,22,35)(18,28,30,33,32,26,45,46,31,47,48,27,44,37,29,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fh,d^2=eh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413282);